1

Resolvability of gravity field parameters

in a repeat orbit: degree/order limit

 

J. Klokočník1, J. Kostelecký2, C. A. Wagner3, A. Bezděk1

 1Astronomical Institute, Academy of Sciences of the Czech Republic, p.r.i. (ASÚ)

CZ – 251 65 Ondřejov Observatory, Czech Republic

jklokocn@asu.cas.cz, bezdek@asu.cas.cz2Research Institute of Geodesy, Topography and Cartography (VÚGTK),

CZ – 250 66 Zdiby and TU Ostrava, CZ – 708 33 Ostrava-Poruba, Czech Republic, kost@fsv.cvut.cz3 2801 Roesh Way, Vienna Va., 22181, USA; e-mail: carl.wagner2@verizon.net

IUGG Prague 29 June G2

2

Abstract

One of the limiting factors in the determination of gravity field parameters is the spatial sampling, namely during phases when the satellite is in a repeat orbit at a low order resonance. This often happens when it is freely passing (drifting) through the atmosphere and encountering various repeat orbits or when it is not drifting but is placed into a preselected repeat orbit to perform specific measurements. This research was triggered in 2004 by the significant but only temporary (2–3 months long) decrease of the accuracy of monthly solutions for the gravity field variations derived from GRACE. The reason for the dip was the 61/4 resonance in the GRACE orbits in autumn 2004. At this resonance, the ground track density decreased and large (mainly longitude) gaps appeared in the data-coverage of the globe. The problem of spatial sampling has been studied repeatedly and simple rules have been derived to limit the maximum order for unconstrained solutions (inversions) for gravity field parameters or their variations. We extend this insight over all achievable latitudes and investigate the ground track density and maximum distances between subsatellite points at arbitrary latitude (specifically for CHAMP, GRACE, and resonant tuned GOCE]. We demonstrate clearly how latitude is important and affects the choice of an order resolution limit. A new order resolution rule is presented, based on the average maximum distance between subsatellite points integrated over achievable latitudes.

 Abstract

3

Orbital resonances

Orbital resonance β:α (or β/α)

β nodal revolutions and α nodal days ERM (exact repeat mission, orbit, resonant orbit)

Modelling of gravity field

Dense enough grid of groundtracks

Unique determination of the gravity field theoretically only toβ/2 or β …. (β –α) depending on parity

Search for better RULE for

maximum order of harmonics;

the limit depends also on

latitude and inclination!

4

5

GRACE 61/4

6

6161/461

61/4resonance

7

GRACE 107/7

8

Ground track density in vicinity of 107/7 repeat orbit

9

Ground track density in vicinity of 46/3 repeat orbit

10

Ground track density in vicinity of 31/2 repeat orbit

11

12

13

Orbital resonances and quality of gravity field models

CHAMP: 46/3, 31/2 (3 times), 47/3 (twice),… orbit manoeuvres

GRACE: 61/4, 107/7, 47/3, 31/2,… free fall,

Worse quality of monthly gravity solutions

GOCE: 16/1 Stable orbit altitude during measuring phasesAvoiding gradiometer

measurements near the 16/1 repeat orbit Sophisticated orbit choice and fine orbit tuning is needed

14

GOCE decreased its initial altitude of 280 km to the planned measurement altitude of 254.9 km by means of free fall

GOCE groundtrack animation for the initial free-fall phase

15

61:979 @254.9 km mean altitude, held for about 3 years of operations 62:997 @ 246 km mean altitude, held for 1 cycle in Sep/Oct 2012 67:1079 @ 240 km mean altitude, held for 1 cycle in Dec2012/Jan2013 73:1177 @ 235 km mean altitude, held for 1 cycle in March/April 2013 143:2311 @ 223.9 km mean altitude, starting from early June 2013

(Priv. commun., R. Floberghagen, 2014)

GOCE MOPs

16

GRACE low order orbit resonances encountered

17

Summary from previous ground track plots and the 3D plot of actual distances (densities, spacing) for GRACE (in time evolution):

DENSITY DEPENDS STRONGLY ON LATITUDE(in addition to dependence on β)

Any rule for maximum degree/order of determinable harmonics for gravity field parameters or their time variations

has to account for an effect of latitude!

18

 

19

 

20

Spherical geometry for the derivation of the distance between the ascending and descending tracks at latitude φ:

 

Step 3:

d1 = mod (dda, Δαasc), d2 = Δα

asc - d1, dmax = max (d1, d2).

21

Theoretical max. distances of ground tracks of GRACE computed by our new formula accounting for latitude

dmax

22

A comparison between our theory and data (case of GRACE)

theory

data (actual orbit)

23

The new rule accounting also for latitude

Theoretical distances dmax of the ground tracks for GRACE for several important resonances.

How to estimate a mean value of dmax over all latitudes covered by the given orbit for the givenresonance?

 

24

2. AMDmax from pole to pole is:

AMDmax =IMDmax/π = 4R/β and half this for AMDmin: AMDmin=IMDmin/π = 2R/β.

3. The two extremes of Mmax, β and β/2, suggest the simple formulae for Mmax as a linear function of AMD (note the negative ratio Mmax/AMD forecast above):   Mmax = (β/2) [3 – β AMD/(2R)].  When AMD = AMDmin=2R/β, Mmax = β (one extreme for Nyquist pole to pole coverage). Similarly, when AMD = AMDmax=4R/β, Mmax = β/2 (the other extreme for Nyquist coverage).

Next figure shows Mmax for selected resonances of CHAMP and GRACE.

25

Filled symbols have (alpha, beta) of the same parity, Klokocnik et al, ASR 2015, publ. on line

26

Strong resonance 31/2 in the official GRACE team products

The resonance 31/2 caused problems to nominal Level-2 GRACE products for February 2015. This problem also affected the January and March 2015 solutions. For example, the Feb 2014 monthly gravity fields produced by CSR are limited to:    - A standard 60x60 solution where more aggressive than normal post-processing may be required.    - Additionally, a deviant 60x30 product for experimental use is provided too.    - No 96x96 solution will be generated and delivered.

(Source: GRACE SDS Newsletter for February 2015)

27

Conclusion

Previous rules for maximum order to which gravity field parameters or their variations should be solved (1) β/2 Wagner et al (2006) or (2) β Colombo’s (Nyquist-type) rule or 3) (β/2 or β) when (β –α) is even/odd, respectively (Weigelt et al, 2009). did not account for latitude. But the resolvability limit strongly depends on LATITUDE (see theory above). This was ignored by all old rules.

The new rule Mmax = (β/2) [3 – β AMD/(2R)] accounts also for latitude (via AMD).

References:Colombo O (1984b) The global mapping of gravity with two satellites, in: Netherl Geod Comm, publ on Geodesy 7(3). Klokočník J, Wagner CA, Kostelecký J, Bezděk A, Novák P, McAdoo D (2008) Variations in the accuracy of gravity recovery due to ground track variability: GRACE, CHAMP, and GOCE, J Geod 82, 917–927, doi: 10.1007/s00190-008-0222-0.Wagner CA, Mc adoo D, Klokočník J, Kostelecký J. (2006) Degradation of geopotential recovery from short repeat-cycle orbits: application to GRACE monthly fields, J Geod 80, 94-103, doi: 10.1007/s00190-006-0036-x.Weigelt M, Sideris MG, Sneeuw N (2009) On the influence of the ground tracks on the gravity field recovery from high-low satellite-to-satellite tracking missions: CHAMP monthly gravity field recovery using the energy balance approach revisited, J Geod. 83, 1131–1143, doi:10.1007/s00190-009-0330-5.